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We consider the motion of a charged particle under the electromagnetic field generated by an electrically neutral infinite straight wire with a time-periodic oscillating (AC-DC) current. By using global continuation and topological degree, we identify a bi-parametric family of radially periodic motions. The proofs involve some delicate estimations of the induced electromagnetic field, which can be of independent interest.

In this paper, we shall prove the stability of the particle advection around a fixed vortex in a two-dimensional ideal fluid under the action of a periodic background flow induced by a polynomial f...

We are grateful to Rafael Ortega for bringing to our attention the position-energy system that has been crucial for the regularization of collisions. We also thank the anonymous referees for their valuable comments and suggestions that contributed to improve the initial version of the paper. This work has been realized thanks to the Agencia Estatal de Investigaci´on, Spain and Ministerio de Ciencia, Innovaci´on y Universidades, Spain grants MTM2017-82348-C2-1-P and MTM2017-86795-C3-1-P., We prove the existence and multiplicity of periodic solutions of bouncing type for a second-order differential equation with a weak repulsive singularity. Such solutions can be cataloged according to the minimal period and the number of elastic collisions with the singularity in each period. The proof relies on the Poincaré–Birkhoff Theorem., Agencia Estatal de Investigación, Spain and Ministerio de Ciencia, Innovación y Universidades, Spain grants MTM2017-82348-C2-1-P and MTM2017-86795-C3-1-P

We consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x ″ + f ( t , x ) = 0 . The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems. We show how a rotation number approach, together with the Poincare–Birkhoff theorem and the phase-plane analysis of the spiral properties, allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity. These systems may also be resonant. In particular, our approach can be used to deal with the problems without both global existence of the Cauchy problems associated to the equation and the sign assumption on f. The typical example is a partially superlinear second order equation. Our main result generalizes some classical results of Jacobowitz and Hartman, among others.

Made available in DSpace on 2018-11-26T16:32:48Z (GMT). No. of bitstreams: 0 Previous issue date: 2016-07-15 MINECO Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) MINECO/FEDER Generalitat de Catalunya European Community The Lienard equation x + f (x)x' + g(x) = 0 appears as a model in many problems of science and engineering. Since the first half of the 20th century, many papers have appeared providing existence and uniqueness conditions for limit cycles of Lienard equations. In this paper we extend some of these results for the case of the generalized phi-Laplacian Lienard equation, (phi(x'))' f(x)psi(x') + g(x) = 0. This generalization appears when derivations of the equation different from the classical one are considered. In particular, the relativistic van der Pol equation, (x'/root 1 - (x'/c)(2))' + mu(x(2) - 1)x' + x = 0, has a unique periodic orbit when mu = 0. (C) 2016 Elsevier Inc. All rights reserved. Univ Estadual Paulista, Dept Matemat, Rua Cristovao Colombo 2265, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil Univ Autonoma Barcelona, Dept Matemat, Edifici C, E-08193 Barcelona, Spain Univ Granada, Dept Matemat Aplicada, E-18071 Granada, Spain Univ Estadual Paulista, Dept Matemat, Rua Cristovao Colombo 2265, BR-15054000 Sao Jose Do Rio Preto, SP, Brazil MINECO: MTM2013-40998-P CAPES: 1271113 MINECO/FEDER: UNAB13-4E-1604 MINECO/FEDER: MTM2014-52232-P MINECO/FEDER: FQM-1861 Generalitat de Catalunya: 2014SGR568 European Community: FP7-PEOPLE-2012-IRSES-318999 European Community: FP7-PEOPLE-2012-IRSES-316338

We provide sufficient conditions for the existence of periodic solutions of the Lorentz force equation, which models the motion of a charged particle under the action of an electromagnetic fields. The basic assumptions cover relevant models with singularities like Coulomb-like electric potentials or the magnetic dipole.

In this paper we study the existence and multiplicity of solutions of the Lorentz force equation $$\left(\frac{q'}{\sqrt{1-|q'|^2}}\right)'=E(t,q) + q'\times B(t,q)$$ with periodic or Dirichlet boundary conditions. In Special Relativity, this equation models the motion of a slowly accelerated electron under the influence of an electric field E and a magnetic field B. We provide a rigourous critical point theory by showing that the solutions are the critical points in the Szulkin’s sense of the corresponding Poincare non-smooth Lagrangian action. By using a novel minimax principle, we prove a variety of existence and multiplicity results. Based on the associated Planck relativistic Hamiltonian, an alternative result is proved for the periodic case by means of a minimax theorem for strongly indefinite functionals due to Felmer.

In this paper we introduce new Lusternik–Schnirelman type methods for nonsmooth functionals including the action integral associated to the relativistic Lagrangian of a test particle under the action of an electromagnetic field $$\begin{aligned} {\mathcal {L}}(t,q,q')=1-\sqrt{1-|q'|^2}+q'\cdot W(t,q) - V(t,q), \end{aligned}$$where $$V:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}$$ and $$W:[0,T]\times {\mathbb {R}}^3\rightarrow {\mathbb {R}}^3$$ are two $$C^1$$-functions with V even and W odd in the second variable. By applying them, we obtain various multiplicity results concerning T-periodic solutions of the relativistic Lorentz force equation in Special Relativity, $$\begin{aligned} \left( \frac{q'}{\sqrt{1-|q'|^2}}\right) '=E(t,q) + q'\times B(t,q), \end{aligned}$$where $$ E=-\nabla _q V-\frac{\partial W}{\partial t}, B=\hbox {curl}_q\, W. $$ The zero Dirichlet boundary value conditions are considered as well.

We study the existence of spacelike graphs for the prescribed mean curvature equation in the Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime. By using a conformal change of variable, this problem is translated into an equivalent problem in the Lorentz–Minkowski spacetime. Then, by using Rabinowitz's global bifurcation method, we obtain the existence and multiplicity of positive solutions for this equation with 0-Dirichlet boundary condition on a ball. Moreover, the global structure of the positive solution set is studied.

In this paper we investigate the existence of rotationally symmetric entire graphs (resp. entire spacelike graphs) with prescribed k-th mean curvature function in Euclidean space R n + 1 (resp. Minkowski spacetime L n + 1 ). As a previous step, we analyze the associated homogeneous Dirichlet problem on a ball, which is not elliptic for k > 1 , and then we prove that it is possible to extend the solutions. Moreover, a sufficient condition for uniqueness is given in both cases.

We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime div (a del u/root 1-a(2)vertical bar del u vertical bar(2)) + g (del u, del a)/root 1-a(2)vertical bar del u vertical bar(2) = lambda NH with 0-Dirichlet boundary condition on the unit ball. According to the behavior of H near 0, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs., National Natural Science Foundation of China (NSFC) 11871129, Xinghai Youqing funds from Dalian University of Technology, Spanish MINECO, European Union (EU) MTM2016-78807-C2-1-P MTM2017-82348-C2-1-P

We identify a family of Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes such that the radially symmetric prescribed curvature problem with Neumann boundary condition is solvable on a ball of small radius. Such family contains some examples of interest in Cosmology.

We study the existence of twist periodic solutions of second order differential equations with an attractive–repulsive singularity. Such twist periodic solutions are stable in the sense of Lyapunov. The proof is based on the third order approximation method in combination with some location information obtained by the averaging method and the method of upper and lower solutions on the reversed order.

Using the only admissible rank-two realisations of the Lie algebra of the affine group in one dimension in terms of the Lie algebra of Lie symmetries of the Ermakov–Pinney (EP) equation, some classes of second order nonlinear ordinary differential equations solvable by reduction method are constructed. One class includes the standard EP equation as a special case. A new EP equation with a perturbed potential but admitting the same solution formula as EP itself arises. The solution of the dissipative EP equation is also discussed.

In this paper, we are concerned with periodic solutions, quasi-periodic solutions and unbounded solutions for radially symmetric systems with singularities at resonance, which are 2π-periodic in time. The method is based on the qualitative analysis of Poincare map with action-angle variables. The existence of infinitely many periodic and quasi-periodic solutions or unbounded motions depends on the oscillatory properties of a certain function.

For the Gylden-Meshcherskii-type problem with a periodically cha-nging gravitational parameter, we prove the existence of radially periodic solutions with high angular momentum, which are Lyapunov stable in the radial direction.

Motivated by a recent series of papers by K. Li and co-workers [14] , [15] , [16] , [17] , [18] , we consider the problem of the existence and multiplicity of solutions for the Neumann or periodic BVPs associated to a class of scalar equations of the form x ″ + f ( t , x ) = 0 . The class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems, one considered near zero and the other near infinity. We show how a rotation number approach, together with the Poincare–Birkhoff theorem and a recent variant of it, allows to obtain multiplicity results in terms of the gap between the Morse indexes of the referred linear systems at zero and at infinity. These systems may also be resonant. When the gaps are sufficiently large, our multiplicity results improve the ones obtained by variational methods in the quoted papers. Also, our approach allows a description of the solutions obtained in terms of their nodal properties.

We study the Dirichlet problem with mean curvature operator in Minkowski space div ( ∇ v 1 − | ∇ v | 2 ) + λ [ μ ( | x | ) v q ] = 0 in B ( R ) , v = 0 on ∂ B ( R ) , where λ > 0 is a parameter, q > 1 , R > 0 , μ : [ 0 , ∞ ) → R is continuous, strictly positive on ( 0 , ∞ ) and B ( R ) = { x ∈ R N : | x | R } . Using upper and lower solutions and Leray–Schauder degree type arguments, we prove that there exists Λ > 0 such that the problem has zero, at least one or at least two positive radial solutions according to λ ∈ ( 0 , Λ ) , λ = Λ or λ > Λ . Moreover, Λ is strictly decreasing with respect to R.

It is shown that a periodically forced impact oscillator may exhibit chaotic dynamics on two symbols, as well as an infinity of periodic solutions. Two cases are considered, depending on if the impact velocity is finite or infinite. In the second case, the Poincare map is well defined by continuation of the energy. The proof combines the study of phase-plane curves together with the “stretching-along-paths” notion.

We analyze the non-degeneracy of the linear $2n$-order differential equation $u^{(2n)}+\sum\limits_{m=1}^{2n-1}a_{m}u^{(m)}=q(t)u$ with potential $q(t)\in L^p(\mathbb{R}/T\mathbb{Z})$, by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.

In this paper we study the existence, multiplicity and stability of T-periodic solutions for the equation $\left(\phi(x')\right)'+c\, x'+g(x)=e(t)+s.$

In this paper, we continue the study of the periodic problem for the second-order equation u ′ ′ + f ( u ) u ′ + g ( u ) = h ( t , u ) , where h is a Caratheodory function and f , g are continuous functions on ( 0 , + ∞ ) which may have singularities at zero. Both attractive and repulsive singularities are considered. The method relies on a novel technique of construction of lower and upper functions. As an application, we obtain new sufficient conditions for the existence of periodic solutions to the Rayleigh–Plesset equation.

Motivated by recent experiments studying the dynamics of configurations bearing a small number of vortices in atomic Bose-Einstein condensates (BECs), we illustrate that such systems can be accurately described by ordinary differential equations (ODEs) incorporating the precession and interaction dynamics of vortices in harmonic traps. This dynamics is tackled in detail at the ODE level, both for the simpler case of equal charge vortices, and for the more complicated (yet also experimentally relevant) case of opposite charge vortices. In the former case, we identify the dynamics as being chiefly quasi-periodic (although potentially periodic), while in the latter, irregular dynamics may ensue when suitable external drive of the BEC cloud is also considered. Our analytical findings are corroborated by numerical computations of the reduced ODE system.

We prove new results about the number of isolated periodic solutions of a first order differential equation with a polynomial nonlinearity. Such results are applied to bound the number of limit cycles of a family of planar polynomial vector fields which generalize the so-called rigid systems.

Sufficient conditions for the existence of a solution to the problem u ″ ( t ) = g ( t ) u μ ( t ) − h ( t ) u λ ( t ) + f ( t ) for a.e. t ∈ [ 0 , ω ] , u ( 0 ) = u ( ω ) , u ′ ( 0 ) = u ′ ( ω ) are established.

The stability of the motion of an oscillator with a periodically time-varying mass is under consideration. The key idea is that an adequate change of variables leads to a newtonian equation, where classical stability techniques can be applied: Floquet theory for the linear oscillator, KAM method in the nonlinear case. To illustrate this general idea, first we have generalized the results of [W.T. van Horssen, A.K. Abramian, Hartono, On the free vibrations of an oscillator with a periodically time-varying mass, J. Sound Vibration 298 (2006) 1166–1172] to the forced case; second, for a weakly forced Duffing’s oscillator with variable mass, the stability in the nonlinear sense is proved by showing that the first twist coefficient is not zero.

We study the existence of solution for the one-dimensional $\phi$-laplacian equation $(\phi(u'))'=\lambda f(t,u,u')$ with Dirichlet or mixed boundary conditions. Under general conditions, an explicit estimate $\lambda_0$ is given such that the problem possesses a solution for any $|\lambda

In this paper, we establish two different existence results of positive periodic solutions for second order non-autonomous singular dynamical systems. The first one is based on a nonlinear alternative principle of Leray–Schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity. The second one is based on Schauder's fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions. Recent results in the literature are generalized and significantly improved.

We find new criteria for the existence of closed solutions in a first order polynomial differential equation which contains the Abel equation as a particular case. Such results are applied to the problem of the existence of limit cycles in planar polynomial vector fields.

In a periodically forced semilinear differential equation with a singular nonlinearity, a weak force condition enables the achievement of new existence criteria through a basic application of Schauder's fixed point theorem. The originality of the arguments relies in that, contrary to the customary situation in the available references, a weak singularity facilitates the arguments of the proofs.

Th ee xistence of non-trivial periodic solutions of a general family of second order differential equations whose main model is a Hill’s equation with a cubic nonlinear term arising in different physical applications is proved. c

We investigate the existence and multiplicity of nontrivial periodic bouncing solutions for linear and asymptotically linear impact oscillators by applying a generalized version of the Poincare--Birkhoff theorem to an adequate Poincare section called the successor map. The main theorem includes a generalization of a related result by Bonheure and Fabry and provides a sufficient condition for the existence of periodic bouncing solutions for Hill's equation with obstacle at $x\not =0$.

In this paper, we will give, for the periodic solution of the scalar Newtonian equation, some twist criteria which can deal with the fourth order resonant case. These are established by developing some new estimates for the periodic solution of the Ermakov–Pinney equation, for which the associated Hill equation may across the fourth order resonances. As a concrete example, the least amplitude periodic solution of the forced pendulum is proved to be twist even when the frequency acroses the fourth order resonances. This improves the results in Lei et al. (2003). Twist character of the least amplitude periodic solution of the forced pendulm. SIAM J. Math. Anal. 35, 844–867.

Motivated by the Lazer–Solimini equation and the Brillouin equation, which present a repulsive singularity at the origin, we will develop in this paper some criterion for the (positive) periodic solution to be of twist type. As an application of the criterion, we will give also some quantitative estimates to the region of parameters so that the equations have twist periodic solutions. These, together with the Moser Twist Theorem, imply that the equations have rich dynamics in the neighborhood of the twist periodic solutions. ? 2003 Elsevier Ltd. All rights reserved.

This paper is devoted to study the existence of periodic solutions of the second-order equation x 00 ¼ f ðt; xÞ; where f is a Caratheodory function, by combining some new properties of Green's function together with Krasnoselskii fixed point theorem on compression and expansion of cones. As applications, we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity. In this latter case, our results cover equations with weak singularities and are compared with some recent results by I. Rachunkova´ , M. Tvrdyand I. Vrkoc˘ . r 2002 Elsevier Science (USA). All rights reserved.

A new stability criterion is proved for second-order differential equations with symmetries in terms of the coefficients of the expansion of the nonlinearity up to the third order. Such a criterion provides solutions of twist type, which are Lyapunov-stable solutions with interesting dynamical properties. This result is connected with the existence of upper and lower solutions of a Dirichlet problem and applied to a known equation which model the planar oscillations of a satellite in an elliptic orbit, giving an explicit region of parameters for which there exists a Lyapunov-stable solution.

This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber \Delta^2 u = (-1)^k S_k[u] + \lambda f(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = \Delta u = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k-$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f \in L^1(B_1(0))$ while $\lambda \in \mathbb{R}$. We prove the existence of a Carath\'eodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided $|\lambda|$ is small enough. Moreover, we prove that the solvability set of $\lambda$ is finite, giving an explicity bound of the extreme value.

We study Lyapunov stability for a given equation modelling the motion of an earth satellite. The proof combines bilateral bounds of the solution with the theory of twist solutions.

where c G 0, a ) 0, and p is a continuous T-periodic function for some T ) 0. We are interested in the existence and stability of positive TŽ . periodic solutions of 1 . The existence of T-periodic solutions of this class of equations where the restoring force is a singular nonlinearity that becomes infinite in zero w x has been proved by Lazer and Solimini 2 for the case without friction Ž . w x c s 0 and by Habets and Sanchez 1 for the damped case. However, the stability properties of these solutions have been less studied. It is well Ž Ž . . known that for the autonomous case p t ' p ) 0 , this equation has a 0 unique saddle point. In this paper we prove that the dynamics of the Ž . periodic equation 1 is similar to the autonomous case. For our study is n w x fundamental a Massera’s convergence theorem in R given by Smith in 6 . The paper is divided in three sections. In Section 1, the main results are Ž . stated. It is seen that 1 has a unique unstable periodic solution w.

The first author is partially supported by a GENIL grant YTR-2011-7 (Spain) and by the grant PN-II-RU-TE-2011-3-0157 (Romania). The second author is partially supported by the grant PN-II-RU-TE-2011-3-0157 (Romania). The third author is partially supported by Ministerio de Economia y Competitividad, Spain, project MTM2011-23652., In this paper, by using Leray-Schauder degree arguments and critical point theory for convex, lower semicontinuous perturbations of C1-functionals, we obtain existence of classical positive radial solutions for Dirichlet problems of type div ( √1 − |∇ ∇v v|2 ) + f(|x|; v) = 0 in B(R); v = 0 on @B(R): Here, B(R) = {x ∈ RN : |x| < R} and f : [0; R] × [0; α) → R is a continuous function, which is positive on (0; R] × (0; α), GENIL (Spain) YTR-2011-7, Ministerio de Economia y Competitividad, Spain MTM2011-23652, PN-II-RU-TE-2011-3-0157

The dynamics of one species chemical kinetics is studied. Chemical reactions are modelled by means of continuous time Markov processes whose probability distribution obeys a suitable master equation. A large deviation theory is formally introduced, which allows developing a Hamiltonian dynamical system able to describe the system dynamics. Using this technique we are able to show that the intrinsic fluctuations, originated in the discrete character of the reagents, may sustain oscillations and chaotic trajectories which are impossible when these fluctuations are disregarded. An important point is that oscillations and chaos appear in systems whose mean-field dynamics has too low a dimensionality for showing such a behavior. In this sense these phenomena are purely induced by noise, which does not limit itself to shifting a bifurcation threshold. On the other hand, they are large deviations of a short transient nature which typically appear only after long waiting times. We also discuss the implications of our results in understanding extinction events in population dynamics models expressed by means of stoichiometric relations, This work was partially supported by the MICINN (Spain) through Project MTM2008-02502

where h :R → RN is a continuous and T -periodic function, v is a vector to be 5xed, ; are real numbers and A is a symmetric N × N matrix. Given u∈RN ; we write u= u+ − u−; where u+ denotes the vector whose components are the positive parts of those of u; and similarly for u−: This kind of systems appear, e.g. after discretization in space of partial di0erential equations like the wave or beam equations. They also provide a mathematical model of a system of coupled oscillators with several springs and stops. Let 1; 2; : : : ; N be the eigenvalues of A; in increasing order, and assume i 1 6 2 6 · · ·6 N i : (2)

We provide necessary and sufficient conditions for the existence of T-periodic solutions of a system of second-order ordinary differential equations that models the motion of two or three collinear charged particles of the same sign.